Embedded newform 187.4.g.b.69.2

• Label: 187.4.g.b.69.2
• Level: $187$
• Weight: $4$
• Character: 187.69
• Analytic conductor: $11.033$
• Analytic rank: $0$
• Dimension: $104$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	187.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0333571711\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			69.2
Character	\(\chi\)	\(=\)	187.69
Dual form			187.4.g.b.103.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.87240 + 2.81346i) q^{2} +(1.76969 + 5.44656i) q^{3} +(4.60776 - 14.1812i) q^{4} +(-0.349264 - 0.253755i) q^{5} +(-22.1766 - 16.1123i) q^{6} +(8.05391 - 24.7874i) q^{7} +(10.2223 + 31.4609i) q^{8} +(-4.68969 + 3.40726i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q + 2 q^{2} - 10 q^{3} - 130 q^{4} - 50 q^{5} + 12 q^{6} + 36 q^{7} - 220 q^{8} - 286 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/187\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.87240	+	2.81346i	−1.36910	+	0.994709i	−0.371292	+	0.928516i	\(0.621085\pi\)
							−0.997807	+	0.0661929i	\(0.978915\pi\)
\(3\)	1.76969	+	5.44656i	0.340578	+	1.04819i	0.963909	+	0.266232i	\(0.0857788\pi\)
							−0.623331	+	0.781958i	\(0.714221\pi\)
\(5\)	−0.349264	−	0.253755i	−0.0312391	−	0.0226965i	0.572056	−	0.820214i	\(-0.306146\pi\)
							−0.603295	+	0.797518i	\(0.706146\pi\)
\(7\)	8.05391	−	24.7874i	0.434870	−	1.33839i	−0.458349	−	0.888772i	\(-0.651559\pi\)
							0.893219	−	0.449621i	\(-0.148441\pi\)
\(11\)	27.0757	+	24.4521i	0.742148	+	0.670236i
							\(13\)	−52.1882	+	37.9169i	−1.11341	+	0.808943i	−0.983198	−	0.182543i	\(-0.941567\pi\)
−0.130217	+	0.991486i	\(0.541567\pi\)
\(17\)	13.7533	+	9.99235i	0.196215	+	0.142559i
							\(19\)	26.9379	+	82.9063i	0.325262	+	1.00105i	0.971322	+	0.237767i	\(0.0764155\pi\)
−0.646060	+	0.763286i	\(0.723585\pi\)
\(23\)	197.571			1.79115			0.895574	−	0.444912i	\(-0.146765\pi\)
							0.895574	+	0.444912i	\(0.146765\pi\)
\(29\)	−47.6275	+	146.582i	−0.304972	+	0.938609i	0.674715	+	0.738078i	\(0.264267\pi\)
							−0.979687	+	0.200530i	\(0.935733\pi\)
\(31\)	−96.2946	+	69.9621i	−0.557904	+	0.405341i	−0.830691	−	0.556734i	\(-0.812054\pi\)
							0.272787	+	0.962074i	\(0.412054\pi\)
\(37\)	−107.745	+	331.604i	−0.478733	+	1.47339i	0.362122	+	0.932131i	\(0.382052\pi\)
							−0.840856	+	0.541259i	\(0.817948\pi\)
\(41\)	−82.1323	−	252.777i	−0.312851	−	0.962858i	−0.976630	−	0.214929i	\(-0.931048\pi\)
							0.663778	−	0.747929i	\(-0.268952\pi\)
\(43\)	295.791			1.04901			0.524507	−	0.851406i	\(-0.324250\pi\)
							0.524507	+	0.851406i	\(0.324250\pi\)
\(47\)	40.3626	+	124.223i	0.125266	+	0.385528i	0.993949	−	0.109839i	\(-0.0350335\pi\)
							−0.868684	+	0.495367i	\(0.835034\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.4.g.b.69.2	✓	104
11.2	odd	10		2057.4.a.u.1.5		52
11.4	even	5	inner	187.4.g.b.103.2	yes	104
11.9	even	5		2057.4.a.v.1.48		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.b.69.2	✓	104	1.1	even	1	trivial
187.4.g.b.103.2	yes	104	11.4	even	5	inner
2057.4.a.u.1.5		52	11.2	odd	10
2057.4.a.v.1.48		52	11.9	even	5